Static Analysis of Stochastic Process Algebras Fan Yang Kongens Lyngby 2007 IMM-MSC-2007-9 2 Summary The Performance Evaluation Process Algebra, PEPA, is introduced by Jane Hillston as a stochastic process algebra for modelling distributed systems and especially suitable for performance evaluation. A range of tools has already been developed that apply this algebra to various application areas for different purposes. In this thesis, we present a static analysis more precisely approximating the control structure of processes expressed in PEPA. The analysis technique we adopted is Data Flow Analysis which is often associated with the efficient im- plementation of classical imperative programming languages. We begin the analysis by defining an appropriate transfer function, then with the classical worklist algorithm we construct a finite automaton that captures all possible interactions among processes. With the help of the novel methodology of anno- tating label and layer to the PEPA program, the approximating result is very precise. Later we try to accelerate the analysis by two approaches, and develop algo- rithms for validating the deadlock property of the PEPA program. In addition, the thesis comes out with a tool that fully implements the analyses and it could be used to verify the deadlock property of the PEPA programs in a certain scale. Keywords: PEPA, Date Flow Analysis, Control Structure, Finite Automaton, Deadlock, Static Analysis, Stochastic Process Algebra. ii Preface This thesis is the result of my work at the Informatics and Mathematical Mod- elling department of Technical University of Denmark, for obtaining a M.Sc degree of Computer System Engineering. It corresponds to 30 ECTs points and is carried out in the period through 1st August 2006 to 31st January 2007, under the supervision of Professor Hanne Riis Nielson. Lyngby, January 2007 Fan Yang iv Acknowledgements First of all, I would like to thank Hanne Riis Nielson, my supervisor, for her excellent guidance through out the whole project. I could always get a lot of inspiration from talking with her, which makes the project going pretty smooth and efficient. Then I would like to thank our project partner Stephen Gilmore. He provides me a series of excellent jobshop examples which show preciousness at the last stage of the project. I would also like to thank Raghav Karol, who gave me very useful suggestions on Latex documentation and always try to share his knowledge with me. Then I would like to thank the people at the Language Based Technology group who make me a pleasant stay when I work on the thesis. Lastly, I would like to give special thanks to my beloved girlfriend Ziyan Feng, for her emotional support and proofreading my thesis several times. Of course, I always feel very grateful for the support from my parents, and thank them for trying to make my life easier when I study abroad. vi Contents Summary i Preface iii Acknowledgements v 1 Introduction 1 1.1 Theoretical Background . 2 1.2 Our work . 3 1.3 Thesis Organization . 4 2 Performance Evaluation Process Algebra 7 2.1 Syntax . 7 2.2 Semantics . 9 2.3 The introduction of Label and Layer to PEPA . 11 viii CONTENTS 3 Exposed Actions 15 3.1 Extended Multiset M and Extra Extended Multiset Mex ..... 15 3.2 Calculating Exposed Actions . 19 3.3 Termination . 21 4 Transfer Functions 23 4.1 Extended Multimap T and Extra Extended Multimap Tex .... 24 4.2 Generated Actions . 24 4.3 Killed Actions . 29 4.4 The Transfer Function . 36 5 Constructing the Automaton 41 5.1 The worklist algorithm . 43 5.2 The procedure update ........................ 44 5.3 The computation of enabled exposed actions . 52 5.4 Correctness result . 65 5.5 Termination of the worklist Algorithm . 67 6 Accelerate the Analysis 71 6.1 Method 1 . 72 6.2 Method 2 . 75 6.3 Three Approaches of Analysis on Two Examples . 81 6.4 Discussion of method1 and method2 . 85 CONTENTS ix 7 Deadlock Verification 89 7.1 Deadlock of PEPA . 89 7.2 Detect the Deadlock by Our Analysis . 90 7.3 Detect the Deadlock of Jobshop Examples . 92 8 Conclusion 99 8.1 Achievement . 99 8.2 Limitation . 100 8.3 Future Work . 100 A The Syntax of Jobshop 4 - 11 103 B Design and Demonstration of the Tool 111 B.1 The Parser . 113 B.2 The Analyzer . 113 B.3 A Guide for Using the Tool through an Example . 115 B.4 Analysis Results . 116 B.5 Source Code . 122 x CONTENTS Chapter 1 Introduction In computer science, the process calculi (or process algebras) are a diverse family of related approaches to formally modelling concurrent systems. They provide a tool for the high-level description of interactions, communications, and synchronizations between a collection of independent processes [35]. The most famous ones include: Communicating Sequential Processes(CSP)[24] ,Cal- culus of Communicating Systems (CCS) [26] and Algebra of Communicating Processes(ACP)[9]. Among them there is a branch of process algebras extended with probabilistic or stochastic information, which are usually named stochastic process algebras (SPAs). For example, in CSP tradition, there is Timed CSP [20], in CCS tradi- − tion, there is PEPA [22], and prACPI [8] is in ACP tradition. The SPAs have gained acceptance as one of the techniques available for performance analy- sis. For example, in large computer and communication systems, they could be used to model the system and predict the behavior of a system with respect to dynamic properties such as throughput and response time[23]. However, the system modelling with SPAs always inherits the process algebras’s concurrent essentials and behaves in a complex way. When dynamically execut- ing the system, we sometimes need to be ensured that there must not arise any abnormal event. For example, the system shouldn’t terminate unexpectedly(no deadlock). Furthermore, sometimes even though we are notified that the sys- 2 Introduction tem might go into the deadlock states, we are not satisfied. We are also curious in which steps might we reach those states, because this information would be very useful for understanding the cause of the abnormity and then help people to revise the system to avoid deadlock. In this thesis, we are going to develop a static analysis for one kind of SPAs: Performance Evaluation Process Algebra (PEPA) [22], aiming at answering the following two questions for the system modelled with PEPA: • Does the system potentially have chance to go into the deadlock states? • If there exist deadlock states, how does the system behave before reaching those states? In the following subsections, first we will introduce the theoretical background of our work. Then we give a short description of the real work we accomplished. In the last of this chapter we will outline the structure of the thesis. 1.1 Theoretical Background 1.1.1 Related Works In recently years, there are several methods of applying static analysis techniques to highly concurrent languages and a variety of process algebras. For process algebras, the works are mainly based on three approaches: • One line of work is to adapt Type Systems from the functional and object- oriented languages to express meaningful properties of the process alge- bras(e.g. [27, 34]). • Another line of work is based on Control Flow Analysis. The process alge- bras that have been extensively researched are: pi-calculus [10], variants of mobile ambients(e.g. [12, 29]) and process algebras for cryptographic protocols such as Lysa [11]. • The last line of work just emerges recently, and it uses the classical ap- proach – Data Flow Analysis to focus on analyzing transitions instead of configurations of the models (Configurations are the main concern of the previous two approaches). The process algebras that has already been done includes CCS [15, 16]. 1.2 Our work 3 Our work adopts the last approach, because its special feature of transition tracking would help us easily answer not only the first question (verify deadlock property) but also the second question mentioned above (find paths leading to deadlock states). And the work in [15, 16] inspires our own work quite a lot. 1.1.2 Analysis Techniques Our work is based on the category of program analysis, in particular the Data Flow Analysis, which will be introduced briefly as follows. Program analysis offers static compile-time techniques for predicting safe and computable approximations to the set of values or behaviors arising dynamically at run-time when executing a program on a computer[28]. The safe here means that analysis is based on formal semantics (Our job is based on the semantics of PEPA). The approximations are usually divided into three classes:(a)Over-approximation captures the entire behavior of the pro- gram. (b)Under-approximation captures a subset of all possible behavior of the program in reality.(c)Undecidable-approximation can’t decide whether the the approximation behaviors belong to the program or not. In our work, we will use both (a) and (b) approaches for approximation. The Data Flow Analysis is among four classical program analysis approaches, which are Data Flow Analysis, Constraint Based Analysis, Abstract Interpreta- tion and Type and Effect Systems. In Data Flow Analysis, it is customary to think of a program (written in traditional programming language) as a graph: the nodes are basic blocks and the edges describe how control might pass from one basic block to another. The transfer functions associated with basic blocks are often specified as Bitvector Frameworks or more general as Monotone Frame- works. The transfer functions in Bitvector Frameworks will always remove in- formation no longer appropriate, and at the same time generate appropriate information to the basic blocks which will form new basic block.